Shallow water

here h(x,y, t) denotes the thickness of the water layer at point
(x, y) at time t, H(x,y) is the bottom topography,
(u, v) is the mass-flow of the water layer,
2g is the gravitational acceleration and t,x,y subscripts
denote partial derivatives.
The shallow-water equations are derived from the depth-averaged
incompressible Navier-Stokes equations for the case where the surface
perturbation is much smaller than the typical horizontal length scale.
The simplest possible scheme is the first-order Lax-Friedrichs
scheme
Qx,y+1 = (Qx+1,y +
Qx-1,y + Qx,y+1 + Qx,y-1)/4 +
Δt Sx,y + [F(Qx+1,y) -
F(Qx-1,y)]Δt/2Δx
+ [F(Qx,y+1) - F(Qx,y-1)]Δt/2ΔyThis is a very robust scheme, which unfortunately gives excessive smearing
of nonsmooth parts of the solution.